Projective semilinear group

Definition

Let $V\ne0$ be a finite-dimensional vector space over a field $k$ . The group $\Gamma L(V)$ consists of all bijective semilinear self-maps of $V$ , allowing every automorphism of $k$ . Its projective semilinear group is

\operatorname{P\Gamma L}(V):=\Gamma L(V)/(k^\times I).

The scalar subgroup is normal because a $\sigma$ -semilinear map conjugates $\lambda I$ to $\sigma(\lambda)I$ .

Action and exact sequence

Every semilinear bijection sends lines to lines, so $\operatorname{P\Gamma L}(V)$ acts faithfully on the incidence geometry of $\mathbb P(V)$ when $\dim_kV\ge2$ . The automorphism associated to a nonzero semilinear map is unique, and gives an exact sequence

1\longrightarrow\operatorname{PGL}(V) \longrightarrow\operatorname{P\Gamma L}(V) \longrightarrow\operatorname{Aut}(k) \longrightarrow1.

For $V=k^n$ , choosing a basis lifts $\sigma\in\operatorname{Aut}(k)$ by applying $\sigma$ coordinatewise, so this sequence splits, though the splitting depends on coordinates.

Collineations

In projective dimension at least two, the fundamental theorem of projective geometry identifies $\operatorname{P\Gamma L}(V)$ with the full group of collineations of the Desarguesian projective space $\mathbb P(V)$ . If $\operatorname{Aut}(k)$ is trivial, then $\operatorname{P\Gamma L}(V)=\operatorname{PGL}(V)$ .

References

Emil Artin, Geometric Algebra, Interscience, 1957. Relevant: Chapter II, §§3–4.
Peter J. Cameron, Projective and Polar Spaces, Queen Mary and Westfield College, 1992. Author-maintained text. Relevant: Chapter 1.